9781439819685-f

GearCutting

ToolsFundamentals of Design

and Computation

© 2010 Taylor and Francis Group, LLC

CRC Press is an imprint of theTaylor & Francis Group, an informa business

Boca Raton London New York

GearCutting

Tools

S t e p h e n P. R a d z e v i c h

Fundamentals of Designand Computation


CRC PressTaylor & Francis Group6000 Broken Sound Parkway NW, Suite 300Boca Raton, FL 33487-2742

© 2010 by Taylor and Francis Group, LLCCRC Press is an imprint of Taylor & Francis Group, an Informa business

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International Standard Book Number: 978-1-4398-1967-8 (Hardback)

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Library of Congress Cataloging-in-Publication Data

Radzevich, S. P. (Stepan Pavlovich)Gear cutting tools : fundamentals of design and computation / by Stephen P. Radzevich.p. cm.

Includes bibliographical references and index.ISBN 978-1-4398-1967-81. Gear-cutting machines. 2. Gearing--Design and construction. 3. Metal-cutting tools--Design

and construction--Mathematical models. 4. Machinery, Kinematics of--Mathematical models. I. Title.

TJ187.R34 2010621.9’44--dc22 2009031422

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vii

Contents

Preface ........................................................................................................................................... xixAcknowledgments ...................................................................................................................... xxiIntroduction ............................................................................................................................... xxiiiSyntax ......................................................................................................................................... xxxi

I BasicsSection

1. Gears: Geometry of Tooth Flanks .......................................................................................31.1 Basic Types of Gears .....................................................................................................31.2. Analytical Description of Gear Tooth Flanks ...........................................................6

1.2..1 Tooth Flank of an Involute Spur Gear .........................................................91.2..2. Tooth Flank of an Involute Helical Gear ................................................... 101.2..3 Tooth Flank of a Bevel Gear ........................................................................ 141.2..4 Tooth Flank of a Helical Bevel Gear .......................................................... 15

1.3 Gear Tooth for Surfaces That Allow Sliding ........................................................... 16

2. Principal Kinematics of a Gear Machining Process ...................................................... 192..1 Relative Motions in Gear Machining ....................................................................... 19

2..1.1 Elementary Relative Motions of the Work Gear and the Gear Cutting Tool ................................................................................................... 2.0

2..1.2. Feasible Relative Motions of the Work Gear and the Gear Cutting Tool ................................................................................................... 2.1

2..2.. Rolling of the Conjugate Surfaces ............................................................................2.3

3. Kinematics of Continuously Indexing Methods of Gear Machining Processes .................................................................................................................................2.53.1 Vector Representation of the Gear Machining Mesh ............................................2.53.2. Kinematic Relationships for the Gear Machining Mesh ...................................... 32.3.3 Configuration of the Vectors of Relative Motions .................................................. 37

3.3.1 Principal Features of Configuration of the Rotation Vectors ................. 373.3.2. Classification of Gear Machining Meshes ................................................ 39

3.4 Kinematics of Gear Machining Processes...............................................................42.

4. Elements of Coordinate Systems Transformations .......................................................434.1 Coordinate System Transformation .........................................................................43

4.1.1 Introduction ...................................................................................................434.1.2. Translations ...................................................................................................444.1.3 Rotation about Coordinate Axis .................................................................464.1.4 Resultant Coordinate System Transformation ......................................... 474.1.5 Screw Motion about a Coordinate Axis ....................................................484.1.6 Rolling Motion of a Coordinate System ....................................................504.1.7 Rolling of Two Coordinate Systems ........................................................... 52.


viii Contents

4.2. Conversion of the Coordinate System Orientation ................................................544.3 Direct Transformation of Surfaces Fundamental Forms ......................................55

II Form Gear Cutting ToolsSection

5. Gear Broaching Tools .......................................................................................................... 595.1 Kinematics of the Gear Broaching Process ............................................................. 595.2. Generating Surface of a Gear Broach .......................................................................605.3 Cutting Edges of the Gear Broaching Tools ............................................................ 61

5.3.1 Rake Surface of Finishing Teeth of a Gear Broach ................................... 615.3.2. Clearance Surface of Gear Broach Teeth ...................................................64

5.4 Chip Removal Diagrams ...........................................................................................655.5 Sharpening of Gear Broaches ...................................................................................665.6 A Concept of Precision Gear Broaching Tool for Machining Involute

Gears ............................................................................................................................. 705.7 Application of Gear Broaching Tools .......................................................................72.

5.7.1 Broaching Internal Gears ............................................................................ 735.7.2. Broaching External Gears ............................................................................ 73

5.8 Shear-Speed Cutting .................................................................................................. 745.8.1 Principle of Shear-Speed Cutting of Gears ............................................... 745.8.2. Profiling of Form Tools for Shear-Speed Cutting of Gears ..................... 765.8.3 Application of Shear-Speed Cutting .......................................................... 79

5.9 Rotary Broaches: Slater Tools ....................................................................................805.10 Broaching Bevel Gear Teeth ...................................................................................... 81

5.10.1 Principle of the Revacycle Process of Cutting of Gear Teeth ................. 82.5.10.2. Revacycle Cutting Tools ...............................................................................835.10.3 Profiling of a Cutter for Machining Bevel Gears Using the

Revacycle Process .........................................................................................855.10.4 Application of the Revacycle Process of Cutting of Gear Teeth.............90

6. End-Type Gear Milling Cutters ......................................................................................... 916.1 Kinematics of Gear Cutting with End-Type Milling Cutter ................................. 916.2. Generating Surface of the End-Type Gear Milling Cutter .................................... 92.

6.2..1 Equation for the Generating Surface of an End-Type Milling Cutter for Machining Spur Involute Gears ............................................... 92.

6.2..2. Equation for the Generating Surface of an End-Type Milling Cutter for Machining Helical Involute Gears ........................................... 96

6.2..3 Elements of Intrinsic Geometry of the Generating Surface of End-Type Milling Cutters .......................................................................... 100

6.3 Cutting Edges of the End-Type Gear Milling Cutter ........................................... 1016.3.1 Rake Surface of the Milling Cutter for Machining of

Involute Gears ............................................................................................. 1016.3.2. Clearance Surface of the Milling Cutter for Machining of

Involute Gears ............................................................................................. 1056.3.3 Cutting Edge Geometry of the End-Type Milling Cutter ..................... 108

6.4 Accuracy of Machining of Gear Tooth Flanks with End-Type Milling Cutters ........................................................................................................................ 115


Contents ix

6.4.1 Cusps on Tooth Flanks of Spur Gear ....................................................... 1156.4.2. Cusps on the Tooth Flanks of a Helical Gear ......................................... 117

6.5 Application of Gear Milling Cutters ...................................................................... 119

7. Disk-Type Gear Milling Cutters ...................................................................................... 12.37.1 Kinematics of Gear Cutting with Disk-Type Milling Cutter .............................. 12.37.2. Generating Surface of the Disk-Type Gear Milling Cutter ................................. 12.4

7.2..1 Equation for the Generating Surface of Disk-Type Milling Cutters for Machining Spur Involute Gears ......................................................... 12.5

7.2..2. Equation for the Generating Surface of the Disk-Type Milling Cutter for Machining Helical Involute Gears ......................................... 12.7

7.2..3 Elements of the Intrinsic Geometry of the Generating Surface of Disk-Type Milling Cutters ......................................................................... 130

7.3 Cutting Edges of the Disk-Type Gear Milling Cutter .......................................... 132.7.3.1 Rake Surface of the Milling Cutter for Machining Involute Gears ..... 132.7.3.2. Clearance Surface of the Milling Cutter for Machining

Involute Gears ............................................................................................. 1347.4 Profiling of the Disk-Type Gear Milling Cutters .................................................. 136

7.4.1 Use of the Descriptive Geometry–Based Method of Profiling............. 1367.4.2. Analytical Profiling of Disk-Type Gear Milling Cutters ....................... 138

7.5 Cutting Edge Geometry of the Disk-Type Milling Cutter .................................. 1437.6 Disk-Type Milling Cutters for Roughing of Gears ............................................... 1477.7 Accuracy of Gear Tooth Flanks Machined with Disk-Type Milling Cutters ... 152.7.8 Application of Disk-Type Gear Milling Cutters ................................................... 154

8. Nontraditional Methods of Gear Machining with Form Cutting Tools ................. 1638.1 Plurality of Single Parametric Motions ................................................................. 1638.2. Implementation of the Single Parametric Motions for Designing of Form

Gear Cutting Tool ..................................................................................................... 1668.2..1 End-Type Gear Milling Cutter .................................................................. 1668.2..2. Disk-Type Gear Milling Cutter ................................................................. 1678.2..3 Face Gear Milling Cutter ........................................................................... 1688.2..4 Internal Round Broach for Cutting Spur and Helical Gears ................ 1698.2..5 Internal Round Broach for Machining Straight Bevel Gears ............... 170

8.3 Diversity of Form Tools for Machining a Given Gear ......................................... 172.8.3.1 Machining of an Involute Worm on a Lathe ........................................... 172.8.3.2. Milling of an Involute Worm .................................................................... 1758.3.3 Thread Whirling ......................................................................................... 1778.3.4 Grinding of an Involute Worm ................................................................. 178

8.4 Classification of Form Gear Tools .......................................................................... 181

III Cutting Tools for Gear Generating: Parallel-Axis Section Gear Machining Mesh

9. Rack Cutters for Planing of Gears .................................................................................. 1879.1 Generating Surface of a Rack Cutter ...................................................................... 1879.2. On the Variety of Feasible Tooth Profiles of Rack Cutters .................................. 191


x Contents

9.3 Cutting Edges of the Rack Cutter ........................................................................... 1939.3.1 Rake Surface of a Rack Cutter ................................................................... 1949.3.2. Clearance Surface of a Rack Cutter .......................................................... 195

9.4 Profiling of Rack Cutters ......................................................................................... 1969.4.1 Profiling of Rack Cutters Using DG-Based Methods ............................ 1979.4.2. Analytical Profiling of Rack Cutters ........................................................ 198

9.5 Cutting Edge Geometry of the Rack Cutter .......................................................... 1999.5.1 Computation of the Cutting Edge Geometry for Lateral Cutting

Edges ............................................................................................................2.009.5.2. Possible Improvements in the Geometry of Lateral Cutting

Edges ............................................................................................................ 2.039.6 Chip Thickness Cut by Cutting Edges of the Rack Cutter Tooth ...................... 2.079.7 Accuracy of the Machined Gear ............................................................................. 2.12.

9.7.1 Satisfaction of the Fifth Condition of Proper PSG ................................. 2.12.9.7.2. Satisfaction of the Sixth Condition of Proper PSG ................................. 2.15

9.8 Application of Rack Cutters .................................................................................... 2.189.9 Potential Methods of Gear Cutting and Designs of Rack-Type Gear

Cutting Tools ............................................................................................................. 2.2.0

10. Gear Shaper Cutters I: External Gear Machining Mesh ............................................2.2.310.1 Kinematics of Gear Shaping Operation.................................................................2.2.310.2. Generating Surface of a Gear Shaper Cutter.........................................................2.2.510.3 Cutting Edges of the Shaper Cutter .......................................................................2.2.9

10.3.1 Rake Surface of a Shaper Cutter ...............................................................2.2.910.3.2. Clearance Surface of a Shaper Cutter Tooth ........................................... 2.32.

10.4 Profiling of Gear Shaper Cutters ............................................................................2.3310.5 Critical Distance to the Nominal Cross Section of the Gear Shaper Cutter ....2.3610.6 Cutting Edge Geometry of a Gear Shaper Cutter Tooth ..................................... 2.39

10.6.1 Angle of Inclination of the Lateral Cutting Edge ................................... 2.4010.6.2. Rake Angle of the Lateral Cutting Edge ................................................. 2.4110.6.3 Clearance Angle of the Lateral Cutting Edge ......................................... 2.4310.6.4 Improvement in the Geometry of Lateral Cutting Edges ..................... 2.45

10.7 Desired Corrections to the Gear Shaper Cutter Tooth Profile ........................... 2.4810.8 Thickness of Chip Cut by Gear Shaper Cutter Tooth .......................................... 2.5110.9 Accuracy of Gears Cut with the Gear Shaper Cutter ..........................................2.55

10.9.1 Satisfaction of the Fifth Condition of Proper PSG .................................2.5510.9.2. Satisfaction of the Sixth Condition of Proper PSG .................................2.58

10.10 Application of Gear Shaper Cutters ....................................................................... 2.6010.10.1 Design of Shaper Cutters ........................................................................... 2.6110.10.2. Special Features of the Shaper Cutter Tooth Profile .............................. 2.6310.10.3 Shaper Cutters for Machining of Helical and Herringbone Gears .....2.6410.10.4 Special Designs of Gear Shaper Cutters .................................................. 2.6510.10.5 Typical Gear Shaping Operations ............................................................ 2.7310.10.6 Grinding of Shaper Cutters ....................................................................... 2.74

11. Gear Shaper Cutters II: Internal Gear Machining Mesh ........................................... 2.8111.1 Kinematics of Shaping Operation of an Internal Gear ....................................... 2.8111.2. Design of Shaper Cutters .........................................................................................2.83

11.2..1 Generating Surface of Gear Shaper Cutters ............................................2.83


Contents xi

11.2..2. Profiling of Gear Shaper Cutters ..............................................................2.8311.2..3 Cutting Edge Geometry of Gear Shaper Cutters ...................................2.85

11.3 Thickness of Chip Cut by the Gear Shaper Cutter Tooth ................................... 2.8611.4 Accuracy of Shaped Internal Gears ....................................................................... 2.9011.5 Enveloping Gear Shaper Cutters ............................................................................ 2.9311.6 Application of Gear Shaper Cutters ....................................................................... 2.93

IV Cutting Tools for Gear Generating: Section Intersecting-Axis Gear Machining Mesh

12. Gear Shapers with a Tilted Axis of Rotation ................................................................ 30112..1 Kinematics of Gear Shaper Operation with the Shaper Cutters Having a

Tilted Axis of Rotation ............................................................................................. 30112..2. Determination of the Generating Surface of a Gear Shaper Cutter Having

a Tilted Axis of Rotation ..........................................................................................30412..3 Illustration of Capabilities of the External Intersecting-Axis

Gear Machining Mesh ............................................................................................. 31112..3.1 Shaping of Conical Involute Gears ........................................................... 31112..3.2. Shaping of Face Gears ................................................................................ 311

13. Gear Cutting Tools for Machining Bevel Gears .......................................................... 31513.1 Principal Elements of the Kinematics of Bevel Gear Generation ...................... 31513.2. Geometry of Interacting Tooth Surfaces ............................................................... 317

13.2..1 Principal Elements of the Geometry of the Involute Straight Bevel Gear Tooth Flank ........................................................................................ 318

13.2..2. Generating Surface of the Gear Cutting Tool ......................................... 31913.2..3 Geometry of Tooth Flanks of the Generated Gear................................. 32.3

13.3 Peculiarities of Generation of Straight Bevel Gears with Offset Teeth ............. 32.513.3.1 Generating Surface of the Gear Cutting Tool ......................................... 32.613.3.2. Generating Surface of the Gear Cutting Tool ......................................... 32.6

13.4 Generation of Straight Bevel Gear Teeth ............................................................... 32.813.4.1 Generation of the Plane Ta by Straight Motion of the Cutting Edge ... 32.813.4.2. Machining of Straight Bevel Gears .......................................................... 32.913.4.3 Gear Cutting Tools for Machining Straight Bevel Gears ......................330

13.5 Peculiarities of Straight Bevel Gear Cutting .........................................................33313.6 Milling of Straight Bevel Gears ..............................................................................334

13.6.1 Peculiarities of the Gear Machining Operation .....................................33413.6.2. Design of Milling Cutters ..........................................................................33513.6.3 Specific Features of the Shape of Finished Bevel Gear Flanks ............336

13.7 Machining of Bevel Gears with Curved Teeth ..................................................... 33713.7.1 Peculiarities of the Gear Machining Operation .....................................33813.7.2. Design of Cutters ........................................................................................340

14. Gear Shaper Cutters Having a Tilted Axis of Rotation: Internal Gear Machining Mesh .................................................................................................................34314.1 Principal Kinematics of Internal Gear Machining Mesh ....................................34314.2. Peculiarities of the Gear Cutting Tool Design ......................................................344


xii Contents

14.2..1 Shaping of Internal Gear ...........................................................................34414.2..2. Shaping a Spur Gear with Enveloping Shaper Cutter ...........................34614.2..3 Shaping of External Recessed Tooth Forms with Enveloping

Shaper Cutter ...............................................................................................347

V Cutting Tools for Gear Generating: Spatial Gear Section Machining Mesh

Section V-A Design of Gear Cutting Tools: External Gear Machining Mesh

15. Generating Surface of the Gear Cutting Tool ...............................................................35315.1 Kinematics of External Spatial Gear Machining Mesh .......................................35315.2. Auxiliary Generating Surface of the Gear Cutting Tool ..................................... 35715.3 Examples of Possible Types of Auxiliary Generating Surfaces of Gear

Cutting Tools .............................................................................................................36315.4 Generation of Generating Surface of a Gear Cutting Tool ..................................363

15.4.1 Design Parameters of the Generating Surface of the Gear Cutting Tool .................................................................................................365

15.4.2. Equation of the Generating Surface of the Gear Cutting Tool ............. 37115.4.3 Setting Angle of the Gear Cutting Tool ................................................... 37415.4.4 Complementary Equations ........................................................................ 376

15.5 Use of the DG-Based Methods for Determining the Design Parameters of the Generating Surfaces of the Gear Cutting Tools ............................................. 37815.5.1 Base Helix Angle ψb.c of the Generating Surface of the Gear

Cutting Tool ................................................................................................. 37815.5.2. Base Diameter db.c of the Generating Surface of the Gear Cutting

Tool................................................................................................................38015.6 Possible Types of Generating Surfaces of Gear Cutting Tools ........................... 381

15.6.1 Generating Surface of the Gear Cutting Tool with a Zero Profile Angle ............................................................................................................ 381

15.6.2. Conical Generating Surface of the Gear Cutting Tool ...........................38315.6.3 Generating Surface of a Gear Cutting Tool with an Asymmetric

Tooth Profile ................................................................................................ 38915.6.4 Generating Surfaces of the Gear Cutting Tools Featuring Torus-

Shaped Pitch Surfaces ................................................................................ 39015.7 Constraints on the Design Parameters of the Generating Surface of a Gear

Cutting Tool ............................................................................................................... 392.

16. Hobs for Machining Gears ............................................................................................... 39516.1 Transformation of the Generating Surface into a Workable Gear Cutting

Tool .............................................................................................................................. 39516.2. Geometry and Generation of Rake Surface of a Gear Hob ................................ 399

16.2..1 Geometry of the Rake Surface .................................................................. 39916.2..2. Generation of the Rake Surface ................................................................403

16.2..2..1 Generation of a Rake Surface in the Form of a Plane ...........40316.2..2..2. Generation of a Screw Rake Surface .......................................405


Contents xiii

16.2..2..3 Peculiarities of Generation of a Screw Rake Surface of a Multistart Hob .................................................................... 407

16.2..2..4 Methods for Generation of an Intermittent Rake Surface of the Special-Purpose Gear Hob ..............................409

16.3 Geometry and Generation of Clearance Surfaces of Gear Hobs ....................... 41116.3.1 Equation of the Desired Clearance Surface of the Hob Tooth ............. 41116.3.2. Generation of the Clearance Surface of the Hob Tooth ......................... 415

16.3.2..1 Cutting of the Relieved Clearance Surfaces of the Hob Teeth .................................................................................... 415

16.3.2..2. Grinding of the Relieved Clearance Surfaces of the Hob Teeth ....................................................................................42.3

16.4 Accuracy of Hobs for Machining of Involute Gears ............................................43316.4.1 Preliminary Remarks .................................................................................43316.4.2. Accuracy of an Involute Gear Hob as a Function of Its Design

Parameters ..................................................................................................43516.4.2..1 Analytical Description of the Desired Lateral

Cutting Edge ...............................................................................43616.4.2..2. Analytical Description of the Actual Lateral

Cutting Edge ...............................................................................43616.4.2..3 Machining Surface of an Involute Hob ................................... 43716.4.2..4 Deviation of the Actual Machining Surface from

the Desired Generating Surface of an Involute Hob .............43816.4.3 Impact of Pitch Diameter on the Accuracy of a Gear Hob ...................445

16.4.3.1 Peculiarities of the Relative Motion of the Work Gear and the Hob ................................................................................446

16.4.3.2. Principal Design Parameters of an Involute Hob ..................44916.4.3.3 Elements of Kinematic Geometry of an Involute Hob .........454

16.5 Design of Gear Hobs ................................................................................................ 462.16.5.1 Design Parameters of a Gear Hob ............................................................ 462.16.5.2. Tooth Profile of the Gear Hob ...................................................................46516.5.3 Precision Involute Hobs with Straight Lateral Cutting Edges .............468

16.5.3.1 Principal Design Parameters of the Precision Involute Hob ............................................................................... 470

16.5.3.2. A Method for Resharpening the Precision Involute Hob ....47716.5.3.3 An Involute Hob for Machining Gear with a

Modified Tooth Profile .............................................................. 48116.5.4 Examples of Nonstandard Designs of Involute Hobs ........................... 487

16.5.4.1 Cylindrical Hobs of Nonstandard Design ............................. 48716.5.4.2. Conical Gear Hobs ..................................................................... 49316.5.4.3 Toroidal Gear Hobs .................................................................... 497

16.6 The Cutting Edge Geometry of a Gear Hob Tooth .............................................. 49816.6.1 The Penetration Curve and the Machining Zone in a Gear

Hobbing Operation.....................................................................................50016.6.1.1 Parameters of the G/Hpc Penetration Curve ........................... 50116.6.1.2. Partitioning of the Machining Zone .......................................503

16.6.2. The Cutting Edge Geometry of a Hob Tooth in the Tool-in-Use Reference System ........................................................................................50516.6.2..1 The Tool-in-Use Reference System in a Gear Hobbing

Operation ....................................................................................505


xiv Contents

16.6.2..2. Geometrical Parameters of the Hob Cutting Edge in the Tool-in-Use Reference System ..................................................508

16.6.2..3 The Possibility of Improving a Hob Design on the Premise of the Results of Investigating the Cutting Edge Geometry .................................................................................... 514

16.7 Constraints on the Parameters of Modification of the Hob Tooth Profile ........ 51516.7.1 The Applied Reference Systems ............................................................... 51616.7.2. Kinematics of the Elementary Gear Drive .............................................. 51716.7.3 Computation of the Maximum Allowed Value of the Modification

of the Tooth Profile of an Involute Hob ................................................... 51816.7.4 Normalized Deviation Δm of the Tooth Profile of the Hobbed Gear ...... 52.2.16.7.5 Peculiarities of Involute Hobs with Reduced Addendum .................... 52.416.7.6 Illustrative Examples of the Computation .............................................. 52.7

16.8 Application of Hobs for Machining Gears ............................................................ 52.816.8.1 Peculiarities of a Gear Hobbing Operation ............................................. 52.816.8.2. Cycles of Gear Hobbing Operations ........................................................53416.8.3 Minimum Hob Travel Distance ................................................................ 536

16.8.3.1 Hobbing Time as a Function of the Hob Total Travel Distance ....................................................................................... 536

16.8.3.2. Impact of the Hob’s Idle Distance on the Minimal Neck Width of the Hobbed Cluster Gear ......................................... 537

16.8.3.3 Selection of a Proper Value of the Setting Angle of the Hob ............................................................................................... 538

16.8.3.4 Computation of the Shortest Allowed Hob Idle Distance .......................................................................................540

16.8.3.5 Impact of Tolerance onto the Shortest Possible Hob Idle Distance .......................................................................................545

16.8.3.6 Computation of the Shortest Allowable Approach Distance of the Hob ................................................................... 552.

16.8.3.7 Designing a Hob Featuring a Prescribed Value of the Setting Angle .............................................................................. 555

17. Gear Shaving Cutters ......................................................................................................... 55917.1 Transforming the Generating Surface into a Workable Gear Shaving

Cutter .......................................................................................................................... 55917.1.1 Generating Surface of a Shaving Cutter .................................................. 55917.1.2. Rake Surface of the Cutting Teeth of a Shaving Cutter.........................56017.1.3 Clearance Surface of the Cutting Teeth of a Shaving Cutter ................ 562.17.1.4 Inclination Angle of the Cutting Edges of a Shaving Cutter................ 562.

17.2. Design of the Gear Shaving Cutters ......................................................................56617.2..1 Design Parameters of a Shaving Cutter................................................... 56717.2..2. Serrations on the Tooth Flanks of a Shaving Cutter ..............................56817.2..3 Resharpening of a Shaving Cutter ........................................................... 571

17.3 Axial Method of the Gear Shaving Process .......................................................... 57617.3.1 Kinematics of the Axial Method of the Gear Shaving Process ............ 57617.3.2. Cutting Speed in the Axial Method of Rotary Shaving of

the Gear ........................................................................................................ 57817.3.2..1 Impact of the Crossed-Axis Angle .......................................... 57817.3.2..2. Impact of the Traverse Motion ................................................. 579


Contents xv

17.3.2..3 Impact of Profile Sliding ...........................................................58017.3.2..4 A Resultant Formula for Cutting Speed in Axial Gear

Shaving ........................................................................................ 58717.4 Diagonal Method of the Gear Shaving Process ................................................... 587

17.4.1 Kinematics of the Diagonal Method of the Gear Shaving Process ..........................................................................................................588

17.4.2. Traverse Angle in Diagonal Method of the Rotary Shaving of a Gear............................................................................................................... 589

17.4.3 Cutting Speed in the Diagonal Method of the Rotary Shaving of a Gear............................................................................................................... 590

17.4.4 Optimization of the Kinematics in the Diagonal Method of the Rotary Shaving of a Gear .......................................................................... 592.17.4.4.1 The Concept of the Optimization ............................................ 592.17.4.4.2. Local Topology of the Contacting Tooth Flanks ................... 59517.4.4.3 Applied Coordinate Systems.................................................... 59617.4.4.4 Geometry of Contact of the Tooth Flanks G and T ................ 59817.4.4.5 Optimal Design Parameters of a Shaving Cutter and

Optimal Parameters of the Kinematics of the Rotary Shaving Operation ..................................................................... 602.

17.5 Tangential Method of the Gear Shaving Process .................................................60317.5.1 Kinematics of the Tangential Method of the Gear Shaving

Process ..........................................................................................................60317.5.2. Cutting Speed in the Tangential Method of the Rotary Shaving of

a Gear............................................................................................................60417.5.3 Tangential Shaving of Shoulder Gear: Descriptive

Geometry–Based Approach ......................................................................60517.5.3.1 Maximum Allowed Outer Diameter of a Shaving

Cutter ...........................................................................................60617.5.3.2. Minimum Required Overlap of the Work Gear and the

Shaving Cutter ............................................................................60817.5.3.3 Minimum Required Face Width of a Shaving Cutter ........... 612.

17.5.4 Tangential Shaving of Shoulder Gear: Analytical Approach ............... 612.17.5.4.1 Optimal Design Parameters of a Shaving Cutter .................. 612.17.5.4.2. Influence of the Overlap of a Shaving Cutter over the

Work Gear onto the Accuracy of the Finished Tooth Flanks .......................................................................................... 615

17.6 Plunge Method of the Gear Shaving Process ....................................................... 61917.6.1 Kinematics of the Plunge Method of the Gear Shaving

Process .......................................................................................................... 61917.6.2. Cutting Speed in the Plunge Method of the Rotary Shaving of a

Gear............................................................................................................... 62.017.6.3 Plunge Gear Shaving Process ................................................................... 62.017.6.4 Plunge Shaving of Topologically Modified Gears ................................. 62.1

17.6.4.1 Geometry of a Topologically Modified Gear Tooth Flank ............................................................................................ 62.1

17.6.4.2. Geometry of the Desired Topologically Modified Tooth Flank of a Shaving Cutter ......................................................... 62.4

17.6.4.3 Grinding a Topologically Modified Tooth Flank of the Shaving Cutter ............................................................................ 62.5


xvi Contents

17.6.5 Satisfaction of Conditions of Proper Part Surface Generation When Designing a Shaving Cutter for Plunge Shaving of Gears ............................................................................................................. 62.817.6.5.1 Circular Mapping of Tooth Flanks of a Work Gear and

the Shaving Cutter ..................................................................... 62.917.6.5.2. Shaving Cutter of a Special Design for Plunge Shaving

of Precision Gears ...................................................................... 63117.7 Advances in the Design of the Shaving Cutter ....................................................633

17.7.1 Elements of the Geometry of the Cutting Edges ....................................63317.7.2. Utilization of Features of the Generating Surface of a Shaving

Cutter ............................................................................................................63617.8 Peculiarities of the Gear Shaving Process .............................................................638

17.8.1 Shaving Cutter Selection ........................................................................... 63917.8.2. Requirements for Preshaved Work Gear ................................................. 63917.8.3 Manufacturing Aspects of Gear Shaving Operation ............................64017.8.4 Modification of Tooth Form and Shape ................................................... 64117.8.5 Shaving of Worm Gear .............................................................................. 641

18. Examples of Implementation of the Classification of the Gear Machining Meshes .............................................................................................................64318.1 A Hob for Tangential Gear Hobbing .....................................................................64318.2. A Hob for Plunge Gear Hobbing ............................................................................64418.3 Hobbing of a Face Gear ............................................................................................64518.4 A Worm-Type Gear Cutting Tool with a Continuous Helix-Spiral Cutting

Edge ............................................................................................................................64618.5 Cutting Tools for Scudding Gears ..........................................................................649

18.5.1 Essentials of the Gear Scudding Process ................................................64918.5.2. A Design Concept of a Precision Cutting Tool for the Gear

Scudding Process ........................................................................................64918.5.3 Applications of the Gear Scudding Process............................................ 651

18.6 A Shaper Cutter with a Tilted Axis of Rotation for Shaping Cylindrical Gears ........................................................................................................................... 65118.6.1 The Kinematics of Shaping a Helical Gear with the Straight-Tooth

Shaper Cutter ............................................................................................... 65118.6.2. Principal Elements of Design of the Gear Cutting Tool ........................ 652.18.6.3 A Possible Application for the Gear Shaper Cutter with a Tilted

Axis of Rotation .......................................................................................... 652.18.7 A Gear Cutting Tool for Machining a Worm in the Continuously

Indexing Method ......................................................................................................65318.8 Rack Shaving Cutters ...............................................................................................654

18.8.1 Rack-Type Shaving Cutter .........................................................................65518.8.2. Kinematics of the Rack Shaving Process .................................................655

18.9 A Tool for Gear Reinforcement by Surface Plastic Deformation ....................... 65718.10 Conical Hob for the Palloid Method of Gear Cutting .........................................658

18.10.1 Preamble ...................................................................................................... 65918.10.2. Design of the Conical Hob ........................................................................ 65918.10.3 Kinematics of the Palloid Gear Hobbing Process ..................................66018.10.4 Peculiarities of Design of a Conical Hob for Machining a Work

Gear with Crowned Teeth ........................................................................ 662.


Contents xvii

Section V-B Design of Gear Cutting Tools: Quasi-Planar Gear Machining Mesh

19. Gear Cutting Tools for Machining of Bevel Gears ......................................................66519.1 Design of a Gear Cutting Tool for the Plunge Method of Machining of

Bevel Gears ................................................................................................................66519.1.1 Kinematics ...................................................................................................66519.1.2. Possible Designs of Tools for Machining Bevel Gears .......................... 667

19.2. Face Hob for Cutting Bevel Gear ............................................................................66819.3 More Possibilities for Designing Gear Cutting Tools Based on

Quasi-Planar Gear Machining Meshes ................................................................. 669

Section V-C Design of Gear Cutting Tools: Internal Gear Machining Mesh

20. Gear Cutting Tools with an Enveloping Generating Surface ................................... 6732.0.1 Gear Cutting Tools with a Cylindrical Generating Surface ............................... 673

2.0.1.1 Generating Surface of an Internal Cylindrical Gear Cutting Tool ...... 673

2.0.1.2. Solution to the Inverse Problem of Part Surface Generation ................ 674

2.0.1.3 Examples of Gear Cutting Tools with an Enveloping Cylindrical Generating Surface ..................................................................................... 676

2.0.2. Gear Cutting Tools with a Conical Generating Surface ...................................... 6792.0.2..1 Generating Surface of an Enveloping Conical Gear Cutting Tool ....... 6792.0.2..2. Examples of Gear Cutting Tools with an Enveloping Conical

Generating Surface .....................................................................................6802.0.3 Gear Cutting Tools with a Toroidal Generating Surface ..................................... 681

2.0.3.1 Generating Surface of an Enveloping Toroidal Gear Cutting Tool ..... 681

2.0.3.2. Examples of Gear Cutting Tools with an Enveloping Toroidal Generating Surface .....................................................................................684

21. Gear Cutting Tools for Machining Internal Gears ...................................................... 6872.1.1 Principal Design Parameters of a Gear Cutting Tool for Machining an

Internal Gear ............................................................................................................. 6872.1.1.1 Geometry of an Internal Gear ................................................................... 6872.1.1.2. Kinematics of Machining an Internal Gear ............................................ 6872.1.1.3 Determination of the Generating Surface of a Gear Cutting Tool

for Machining an Internal Gear ............................................................... 6892.1.2. Examples of Gear Cutting Tools for Machining an Internal Gear .................... 689

Conclusion ................................................................................................................................... 693

Appendix A: Engineering Formulae for the Specification of Gear Tooth ..................... 695

Appendix B: Conditions of Proper Part Surface Generation ............................................ 699

Appendix C: Change of Surface Parameters ........................................................................ 703

Appendix D: Cutting Edge Geometry: Definition of the Major Parameters ................. 705

Notation ....................................................................................................................................... 719

References ...................................................................................................................................72.3

Index ............................................................................................................................................. 733


xix

Preface

Gear Cutting Tools: Fundamentals of Design and Computation is intended for mechanical engi-neers and manufacturing engineers who are interested in the scientific basis as well as the practical aspects of the advances in gear machining.

This book is neither a textbook nor a manual. It combines science and practice. The reader will find enough material to understand the geometry of gear cutting tools and the kinematics of a gear machining process.

This book deals with the science of surface generation. The investigation of surface gen-eration process offers an excellent context and motivation for exploring the pervasive ties between geometry and kinematics.

The discussion in the book is based on the DG/K-based approach of surface generation, which is derived from the terms differential geometry (DG) and “kinematics (K) of multipara-metric motion in Euclidean E3 space.” The DG/K approach was developed by the author and disclosed in more detail in some of his previous books ([12.5, 136, 138, 143, 153], etc.).

To the author’s knowledge, this book represents the first attempt to disclose the details of designing optimal gear cutting tools based on the fundamental scientific theory. Most, but not all, important topics in the field of gear cutting tool design are covered in this book. However, this area of gear engineering is too broad, and there remains plenty of room for further improvements in this field.

Special-purpose gear cutting tools for machining noncircular gears are not considered in this monograph. However, the approach discussed in this work can be extended into the area of mechanical engineering as well.


xxi

Acknowledgments

I would like to share the credit for any research success with my numerous doctoral stu-dents with whom I have tested the proposed ideas and applied them in the industry. The contributions of many friends, colleagues, and students in overwhelming numbers cannot be acknowledged individually, and as much as our benefactors have contributed to this work, even their kindness and help must go unrecorded. Thanks are also due to a number of experts, whose suggestions have greatly improved portions of the book.


xxiii

Introduction

“Simplex sigillum veri.” (Simplicity is the seal of truth.)

Latin proverb

This book deals with cutting tools used for machining of gears. Gears are produced in enormous amounts—billions of gears are produced every year. While the automotive industry ranks as the primary consumer of gears, numerous other industries also require huge amounts of gears: construction machinery, agricultural machinery, aerospace indus-try, to name a few.

The gear cutting process is costly. Cost of the gear cutting tools exceeds 50% of the total cost of the gear machining operation. If we place cost savings in the production of every gear in the range of just 10 cents, the total cost savings could reach hundreds of millions of dollars. This constitutes an appealing incentive for engineers and scientists to turn their attention to the gear machining process and the design of gear cutting tools used in the machining of gears.

In writing this book, the author has tried to expand the theory and fill various gaps.The approach used in the text is mainly analytical. However, geometric interpretations

are given, and in places synthetic reasoning is also applied. Furthermore, the author has largely limited himself to the mathematics of algebraic geometry, vector and matrix alge-bra, and elementary calculus. Special mathematical methods are avoided even in the final chapters.

The use of powerful computers makes it reasonable not to derive equations in their final form (as they are often bulky), but just to outline the major steps with which a problem can be solved.

At this point, it is fitting to recall the old Chinese proverb: “The beginning of wisdom is to call things by their right names.” Unfortunately, even among gear specialists there is some ambiguity in the terms used to describe gears and gear-related parameters. In this work, to the extent possible, we follow the conventional terminology. When necessary, definition of new terms is provided.

The development of the scientific classification of kinematic schemes of gear machining process, as well as illustration of the classification with practical designs of gear cutting tools and advanced methods of gear machining are the two reasons, among others, that highly motivated the author to write this text.

The development of the scientific classification of kinematic schemes of gear machining is among the major goals to be disclosed in the book. Identification of the proper place of all known gear cutting tool designs in the developed classification is another important goal of the book. Ultimately, identification of areas where there is plenty of room for further developments in the field of designing optimal gear cutting tools is the third major goal of the book. However, the primary aims of this book are not just limited to these goals.


xxiv Introduction

Historical Background

This study surveys and assesses the considerable body of research on gears and gear machining processes that has accumulated to date. It is not the aim of the author, who has been actively involved in this field of research for more than 30 years, to develop all possible designs of gear cutting tools. That huge task is not within the scope of this text. However, the potential disclosure of all possible conceptual designs of gear cutting tools is another major goal of this book.

Uniqueness of this Publication

This book is unique for many reasons.Most of the material used in this book is new and cannot be found elsewhere.This is the first work in English dedicated solely to the optimal design of gear cutting

tools. The treatment is rigorous and elegant, focusing on the mathematical development of the subject apart from any particular applications. This book is also unique in the sense that most of the material was developed by the author.

The discussion starts with the general concepts and problems, and then specializes gradually to more simple cases. The kinematics of the gear machining mesh is the key point for the proper understanding of the disclosed approach. Eventually it can be under-stood that by just playing with two rotation vectors one can develop a novel method of gear machining as well as a novel gear cutting tool design for this purpose (playing with two rotation vectors, ωg and ωc, along with one or two additional vectors of prime motion, feed motion, etc).

Numerous examples and an extensive bibliography further enhance the usefulness of the book.

Because of its generality and lucid style, this classic work will be invaluable not only to specialists in gear cutting tool design but also to those working in general areas of mechanical/manufacturing engineering.

Intended Audience

Since a perusal of the table of contents may leave the reader wondering as to whether this volume was intended as a textbook, a research monograph, or a historical treatise, some explanatory remarks are perhaps in order. There was, in fact, no conscious intent to aim for any of these—but what has transpired seems to bear the fact it is a combination of all three. Gear experts working in various industries and in academia, as well as university students (senior graduate, graduate, and postgraduate students) are the intended audience of the book.


Introduction xxv

Organization of this Book

For the readers’ convenience, the book is divided into five parts. Each part is composed of about five chapters. Part V, which is large enough, is subdivided into three sections, each of which relates to gear cutting tools designed on the basis of: (a) external gear machining mesh, (b) quasi-planar gear machining mesh, and (c) internal gear machining mesh.

The content of each chapter is briefly outlined below.

Section I. Basics

Fundamental issues of gear cutting tools are covered in this section of the book. The dis-cussion encompasses types of gears to be machined, the analytical representation of the gear tooth flanks, types of relative motions of the work gear, and the gear cutting tool in the gear machining process. The section ends with a brief consideration of the linear trans-formations methods that are practical in designing gear cutting tools. This part of the book is composed of four chapters.

Chapter 1. The geometry of tooth flanks of common types of gears is discussed in this chapter. Along with numerous practical examples of gear designs, equations for the tooth flanks of spur, helical, straight bevel, and helical bevel gears are derived here. The derived equations illustrate the powerful method that can be used to derive an equation for the tooth flank of a gear of any given design. An analytical description of the desired gear tooth flank is a good starting point for the gear cutting tool designer.

Chapter 2. General aspects of the kinematics of gear machining processes are considered in Chapter 2.. The analysis is focused mostly on two issues: (a) possible types of relative motions in the gear machining process, including, but not limited to, elementary relative motions of the work gear and the gear cutting tool, and (b) rolling of the conjugate surfaces over each other. In particular, the readers’ attention is drawn to the practical applications of special types of motions under which a surface allows sliding over itself.

Chapter 3. This chapter focuses on the kinematics of special methods of gear machin-ing processes. The methods are referred to as the continuously indexing methods of gear machining processes. Vector representation of the gear machining mesh and the kine-matic relationships for the gear machining mesh are covered in this chapter. Based on the in-depth analysis of the principal features of configuration of the rotation vectors that specify the gear machining mesh, a scientific classification of types of gear machining meshes is developed. The classification is the key for the development of novel designs of gear cutting tools and new methods of machining gears. Major steps for transforming the kinematics of the gear machining mesh into the corresponding kinematics of the gear machining process are considered.

Chapter 4. Practical methods of the coordinate system transformation are discussed in Chapter 4. Translations and rotations of coordinate systems are considered as particular cases of linear transformations. This chapter illustrates how operators of resultant coordi-nate system transformations can be composed using the operators of elementary coordi-nate system transformations. In addition to conventional operations of coordinate system transformations, new operators of linear transformations are introduced: (a) operator of screw motion about a coordinate axis, (b) operator of rolling motion of a coordinate sys-tem, and (c) operator of rolling of two coordinate systems. Formulae for the conversion of coordinate system orientation as well as methods for direct transformation of surface fundamental forms are also covered in this chapter.


xxvi Introduction

Section II. Form Gear Cutting Tools

Gear cutting tools featuring the generating surface that is congruent to tooth surfaces of the gear to be machined are discussed in this part of the book. The kinematics of the gear machining processes used for the purpose of form gear cutting tools is the simplest pos-sible. However, simple kinematics of surface machining often entails cutting tools of very complex designs.

Chapter 5. Design of cutting tools for broaching gears is the subject of this chapter. The discussion begins with the analysis of the kinematics of the gear broaching process, and is followed by generation of the generating surface of the gear cutting tool, transformation of the generating surface into a workable gear cutting tool, cutting edge geometry, and by chip removal diagrams. Both analytical methods and DG-based methods of analysis are widely used in this chapter. Sharpening of gear broaches is an issue of particular consid-eration. The discussion of gear broaching tools ends with a detailed disclosure of the con-cept of precision gear broaching tools for machining involute gears and with application issues of gear broaching tools.

The concept of broaching is the cornerstone of other methods of gear cutting. These methods include, but not limited to, (a) Shear-Speed cutting of gears, (b) rotary broaching with slater tools, (c) Revacycle method of gear cutting, etc. The principles of gear cutting tool designs for these methods of gear machining are also covered in this chapter.

Chapter 6. Design methods of end-type milling cutters for machining spur and helical gears are discussed in this chapter. At the outset, the kinematics of the gear cutting process with end-type milling cutter is discussed. This is followed by an analytical description of the secondary generating surface of the end-type milling cutter, and by an investigation of the cutting edge geometry. The analysis makes it possible to compute the cutting wedge angles at any point of interest within the cutting edge. Accuracy of the milled gears is dis-cussed from the perspective of the geometry of interacting surfaces and the kinematics of the gear cutting process. Application of the end-type milling cutters is considered at the end of the chapter.

Chapter 7. Disk-type milling cutters as well as their design and analysis are discussed in this chapter. Analysis of the kinematics of the gear milling operation allows for determin-ing the secondary generating surface of the milling cutter. This is followed by an analytical description of the generating surfaces of milling cutters designed for machining spur and helical gears. Next, the intrinsic geometry of the generating surfaces of the milling cutters is investigated. Considered together, an equation of the secondary generating surface of the milling cutter, an equation of the rake surface, and an equation of the clearance surface of the milling cutter tooth allow for the analytical representation of the cutting edge of the cutting tool. Profiling of milling cutters is considered from the perspective of implement-ing both DG-based methods and analytical methods in profiling of form cutting tools. The cutting edge geometry of the disk-type milling cutter is analyzed using elements of vector algebra and calculus. The peculiarities of milling cutter designs for roughing of gears are considered. Accuracy issues are investigated from the perspective of satisfaction/violation of conditions of proper part surface generation. The chapter ends with a discussion of the principal features of the practical application of disk-type milling cutters for cutting spur and helical gears.

Chapter 8. Investigation of the kinematics of gear machining operations makes it possible to develop novel gear machining methods as well as novel gear cutting tool designs. All possible methods of gear cutting are referred to as nontraditional methods of gear machin-ing with form cutting tools.


Introduction xxvii

The consideration begins with the systematic investigation of all possible types of single-parametric motions. Analysis reveals that novel schematics of gear cutting with (a) end-type milling cutters, (b) disk-type milling cutters, (c) face gear milling cutters, (d) with rotary broaches, etc., can be developed. In particular, various methods and cutting tools of various designs intended for machining worms are considered. In closing, a possibility of classifying form gear cutting tools is discussed.

Section III. Cutting Tools for Gear Generating: Parallel-Axis Gear Machining Mesh

Gear cutting tool designs and gear cutting methods featuring parallel axes of the work gear and the gear cutting tool are investigated in this part of the book. This section begins with a discussion of the kinematics of the parallel-axis gear machining mesh, which is the key issue for the proper understanding of the design and operation of gear cutting tools of many designs.

Chapter 9. Rack cutters for planing gears are the simplest example of generating gear cutting tools featuring parallel-axis gear machining mesh. The geometry and kinematics of machining of gears with rack cutters are analyzed. For profiling of rack cutters, use of the graphical method that is based on the wide implementation of DG-based methods and analytical methods is recommended. Methods of vector algebra are used to investigate the cutting edge geometry of rack cutters. Based on the results of the analysis, improvements are made in the geometry of the lateral cutting edges. Chip thickness cut by cutting edges of the rack cutter tooth is analyzed. Accuracy of the tooth flanks of machined gears is analyzed from the perspective of using the approach that is commonly referred to as the kinematic geometry of surface machining. Application of rack cutters as well as potential gear cutting methods and rack-type gear cutting tool designs are discussed in the final two sections.

Chapter 10. Design of gear shaper cutters and methods of shaping of external gears are investigated in Chapter 10. The external parallel-axis gear machining mesh is comple-mented with the cutting motion. In this way, the kinematics of the gear shaping process is derived. Analytical description of the generating surface of the gear shaper cutter, types of the rake surface, and geometry of the clearance surface are discussed. Methods of vector algebra are used to investigate the cutting edge geometry of gear shaper cutters. Valuable improvements in the cutting edge geometry are discovered based on the results of the research. Elements of the kinematics of gear meshing are used for the analytical descrip-tion of thickness of chip cut by the gear shaper cutter tooth, as well as for the analytical description of the deviations of the machined gear tooth surface from the desired geom-etry of the gear tooth flanks. The analysis is focused on verifying whether the fifth and sixth conditions of proper part surface generation are satisfied. Issues relating to the appli-cation of gear shaper cutters cover both conventional and special-purpose designs of gear shaper cutters for machining spur and helical gears. As an example, typical operations of gear shaping are briefly described. Methods of grinding of the rake and clearance surfaces of gear shaper cutters are discussed at the end of the chapter.

Chapter 11. The design of gear cutting tools using the parallel-axis internal gear machin-ing mesh is discussed in this chapter. The gear machining mesh is complemented with the primary motion. Design procedure encompasses (1) generation of the generating surface of the cutting tool, (2.) profiling of the gear shaper cutters, and (3) exploration of the cut-ting edge geometry of the gear shaper cutter. Computation of thickness of chip cut by the gear shaper cutter tooth and accuracy of the shaped internal gears are also covered in this chapter. Because the same parallel-axis internal gear machining mesh is not limited to the


xxviii Introduction

shaping of internal gears, but can also be applied for the design purposes of enveloping shaper cutters for machining external gears, the related issues are briefly discussed in this chapter. Some important issues relating to the practical applications of the gear shaper cut-ters are discussed at the end of the chapter.

Section IV. Cutting Tools for Gear Generating: Intersecting-Axis Gear Machining Mesh

Gear cutting tool designs and gear cutting methods featuring intersecting axes of the work gear and the gear cutting tool are investigated in this part of the book. The discus-sion starts with the investigation of the kinematics of the intersecting-axis gear machining mesh, which is the key issue for the proper understanding of the design and operation of gear cutting tools of many designs.

Chapter 12. Gear shaper cutters with a tilted axis of rotation are examples of gear cutting tools designed based on the principle of the external intersecting axis gear machining mesh. The kinematics of the intersecting-axis gear machining mesh is complemented with an additional primary motion (motion of cut) of the cutting tool to be designed. In this way, the kinematics of the gear machining process using gear shaper cutters with a tilted axis of rotation is derived. For the particular kinematics of gear cutting, the generating surface of the gear shaper cutter with a tilted axis of rotation is derived. Capabilities of the external gear machining mesh are illustrated with several cases of its applications in gear cutting tool design.

Chapter 13. Design of gear cutting tools for machining bevel gears is discussed in this chapter. Gear cutting tools are designed using a round rack that is properly meshing with the gear to be machined. The discussion begins with the analysis of the principal elements of the kinematics of bevel gear generation. Geometry of the interacting surfaces, namely, the involute straight bevel gear flank and the generating surface of the gear cutting tool, along with the geometry of tooth flanks of the generated gears, is analyzed. Generation of straight bevel gears with offset teeth is also covered in this chapter. The discussed geo-metrical and kinematical aspects are used for the comprehensive investigation of methods of planing and milling of straight bevel gears, and milling of bevel gears with curved teeth. The design of gear cutting tools used for this purpose is discussed along with the corresponding methods of gear machining. It is stressed here that most bevel gears are a type of practical approximation to the desired tooth geometry.

Chapter 14. The possibility of designing gear cutting tools based on an internal intersect-ing-axis gear machining mesh is discussed in this chapter. Shaping of internal gears using the gear shaper cutter with a tilted axis of rotation, as well as cutting of external gears with enveloping gear shaper cutters with a tilted axis of rotation are good examples in this concern. Shaping of external recessed tooth forms using an enveloping gear shaper cutter is one more opportunity to utilize an internal intersecting-axis gear machining mesh for the purpose of designing gear cutting tools. However, gear machining meshes of this type have not been subjected to a comprehensive investigation as yet.

Section V. Cutting Tools for Generating of Gears: Spatial Gear Machining Mesh

Most gear cutting tool designs are based on the generating principle. Spatial gear machin-ing mesh is simulated when machining a gear using these gear cutting tools. The huge numbers of known as well as possible designs of gear cutting tools result in the decision


Introduction xxix

to consider them in three sections, each of which is devoted to: (a) external gear machining mesh, (b) quasi-planar gear machining mesh, and (c) internal gear machining mesh.

Section V-A. Design of Gear Cutting Tools: External Gear Machining Mesh

Gear cutting tool designs based on the principle of external spatial gear machining mesh are considered in this section. The section starts with the investigation of the possible types of generating surfaces of the gear cutting tools. Designs of gear hobs, shaver cutters, etc., are considered in consequent chapters.

Chapter 15. The generation principles of the generating surfaces of gear cutting tools are investigated in this chapter. The discussion begins with the analysis of the kinematics of the external spatial gear machining mesh. Possible types of auxiliary generating surfaces together with practical methods of their generation are considered. Generation of the gen-erating surface of the gear cutting tool is based on the possible types of auxiliary generat-ing surfaces. Equations of generating surfaces are derived and design parameters of the surfaces are computed. For the analysis, DG-based methods and analytical methods are used. Next, various types of generating surfaces are constructed. They include, but are not limited to, conventional cylindrical generating surface, cylindrical generating surface with a zero profile angle, conical generating surfaces, generating surfaces featuring an asym-metrical tooth profile, as well as surfaces with a torus-shaped pitch surface. The chapter ends with a brief discussion of the geometrical and kinematical constraints on the design parameters of the generating surface of gear cutting tools.

Chapter 16. Hobs are a perfect example of gear cutting tool design based on the external spatial gear machining mesh. The geometry of the generating surface of gear hobs along with geometry of the rake and clearance surfaces are analyzed. Generation methods for the rake and clearance surfaces of hob teeth are also considered. The derived equations for the working surfaces of the gear hob enable an in-depth analysis of the accuracy of hobs for machining of involute gears. Numerous advanced designs of hobs for machining gears are discussed as examples of implementation. The cutting edge geometry of gear hobs is analytically described in the tool-in-use reference system. To satisfy the necessary condi-tions of proper part surface generation in a gear machining process, constraints on the parameters of modification of the hob tooth profile are investigated. The discussion is fol-lowed by several numerical examples of computation of the design parameters of the hobs. The obtained results can be enhanced for application in the area of designing of hobs for machining noninvolute profiles. Applications of hobs for machining gears are discussed from the perspective of the analytical research on the geometry and kinematics of the gear hobbing process.

Chapter 17. Gear shaving cutters represent another type of gear cutting tool designed on the premise of spatial external gear machining mesh. The procedure of designing a shav-ing cutter begins with the transformation of the generating surface into the workable gear shaving cutter. Peculiarities of the geometry of the rake surface, the clearance surface, and the cutting edge geometry are discussed. Issues on the design parameters of a shaving cut-ter, design features of the serrations, and resharpening of shaving cutters are considered. Four basic methods for gear shaving—(1) axial shaving, (2.) diagonal shaving, (3) tangential shaving, and (4) plunge shaving—are discussed from the perspective of the kinematic geometry of surface machining. The discussion ends with an analysis of the advanced designs of gear shaving cutters and practical issues relating to the implementation of gear shaving processes for shaving precision gears.


xxx Introduction

Chapter 18. Examples illustrating the capabilities of the external crossed-axis gear machining mesh are discussed in this chapter. These examples can be divided into two sections. Design of gear cutting tools and methods for machining cylindrical gears are considered in the first section. Hobs for tangential hobbing, hobs for plunge hobbing, hobs for machining face gears, continuous generating cutting of worms, gear cutting tools for the scudding process, and gear shaper cutters with a tilted axis of rotation for machining cylindrical gears along with rack-type shaper cutters are also discussed in the first section. Design of gear cutting tools and methods for machining conical gears are discussed in the second section. Discussion in this section is limited to special-purpose tools for reinforce-ment of gears by surface plastic deformation and conical hobs for the Palloid method of gear cutting.

Section V-B. Design of Gear Cutting Tools: Quasi-Planar Gear Machining Mesh

Design of gear cutting tools based on the principle of the kinematics of quasi-planar gear machining mesh is discussed in this section. The section begins with a detailed anal-ysis of the vector representation of the kinematics of quasi-planar gear machining mesh.

Chapter 19. This chapter is devoted to cutting tools for machining bevel gears. Gear cut-ting tool designs for novel gear cutting methods are discussed. Topics include cutting tools for plunge cutting of bevel gears, face hobs for continuously indexing method of gear cutting, as well as possible directions for further development.

Section V-C. Design of Gear Cutting Tools: Internal Gear Machining Mesh

This final section of the book deals with designs of gear cutting tools that use internal spatial gear machining mesh. Again, the consideration begins with a detailed analysis of the kinematics of the internal spatial gear machining mesh. For this purpose, vector rep-resentation of internal spatial gear machining mesh is widely used.

Chapter 20. Designs of gear cutting tools with an enveloping generating surface are dis-cussed in this chapter. Gear cutting tools with cylindrical, conical, and toroidal generating surfaces are covered in this chapter. The discussion includes the analytical determination of the generating surface itself, as well as some aspects relating to the accuracy of the machined gears. This is followed by examples of gear cutting tool designs relating to this group of gear cutting tools.

Chapter 21. The kinematics and design of gear cutting tools are briefly discussed in this chapter. The principal design parameters of gear cutting tools are determined. The discus-sion is followed by examples of novel designs of gear cutting tools for machining internal gears.

It took the author years to bring this book to completion.It is inevitable that a study of this nature will lean toward greater emphasis on the

author’s own contributions, if only because they share his perspective on the subject mat-ter. Nevertheless, an effort has been made to summarize the key ideas (if not the techni-cal details) of the most significant developments in the field, and give pointers to many others.

A book of this size is likely to contain omissions and errors. If you have any constructive suggestions, please communicate them to Dr. S. Radzevich ([email protected]).

Stephen P. RadzevichSterling Heights, MI


xxxi

Syntax

Each paragraph is identified first by the chapter number and then by a serial number with the chapter. For example, the second paragraph of chapter 3 is called §3.2.. Any such num-ber inserted suddenly online and within parentheses, for example, (§3.2.), is an invitation to turn to that paragraph for parallel reading. These internal references may be casting either forward to material yet to be encountered, or backward to material that might require a second reading. Appendices appear at the end of the book. They are consequently lettered A, B, C, and D for easy identification.

A list of references appears at the end of the book. These entries are referenced by means of consecutive numbers within square brackets. I have sometimes enlarged this device to refer to specific paragraphs within a quoted work. For example, [1] [§12..13] specifically refers the reader to §12..13 in [1]. The many interconnected themes of the book cannot be reduced, one by one, each to a simple number. Developing as they go, they run interwoven throughout.

No quoted statements or data are made in cases where well-known results are discussed within the text. This is the major reason why the list of references does not contain numer-ous works.

Algebraic symbols are listed separately in the Notation section (p. 719). Many of the drawn figures and much of the written argument are variously based on or bolstered by numerical examples. For the reader interested in reproducing the CAD calculations, most numerical values are given either directly online with the text (but isolated there within square brackets) or, more often, among the Appendices.


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